Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces

Let A be a positive bounded operator on a Hilbert space (H,〈⋅,⋅〉). The semi-inner product 〈x,y〉A:=〈Ax,y〉, x,y∈H, induces a seminorm ‖⋅‖A on H. Let ‖T‖A,wA(T), and cA(T) denote the A-operator seminorm, the A-numerical radius, and the A-Crawford number of an operator T in the semi-Hilbertian space (H,‖⋅‖A), respectively. In this paper, we present some seminorm inequalities and equalities for semi-Hilbertian space operators. More precisely, we give some necessary and sufficient conditions for two orthogonal semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive new upper and lower bounds for the numerical radius of operators in semi-Hilbertian spaces. In particular, we show that116‖TT♯A+T♯AT‖A2+116cA((T2+(T♯A)2)2)≤wA4(T)≤18‖TT♯A+T♯AT‖A2+12wA2(T2), where T♯A is the A-adjoint operator of T. Some applications of the newly obtained inequalities are also provided.